Solve quadratic equations and use continued fractions to find
rational approximations to irrational numbers.

Rational Roots

Stage: 5 Challenge Level:

In this problem you are given that $a$, $b$ and $c$ are natural
numbers. You have to show that if $\sqrt{a}+\sqrt{b}$ is rational
then it is a natural number.

You could use the fact that if $\sqrt{a}+\sqrt{b}$ is rational then
so is its square which means that $\sqrt ab $ is also rational.
Knowing this the next step is to use $$\sqrt{a}(\sqrt{a}+\sqrt{b})
= a+\sqrt{ab}$$ to show that $\sqrt a$ is rational and to do
likewise for $b$.

This is all you need because it has been proved that if $\sqrt a$
is rational then $a$ must be a square number.

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